Structural Catastrophe Operator: A General Framework for Structural Ruptures in Complex Systems

Classical catastrophe theory models discontinuities through singularities in smooth dynamical systems, yet it often fails in domains where structural compatibility is the primary constraint. In this work, we introduce the Structural Catastrophe Operator (SCO), a formal framework for describing transitions between incompatible structural regimes. Within the Theory of Non-Derivability and Admissibility (TNA), a system is defined by a structural predicate S that determines the domain of admissible states. We propose the Bresciano Metric as a formal measure of the structural distance to the admissibility boundary, where a catastrophe occurs as a non-derivable transition S S'. By defining the Structural Catastrophe Operator, we show that regime shifts emerge when internal dynamics can no longer be contained within the existing structural constraint. The framework provides a unified formal language to characterize ruptures across physics, biology, and complex social systems, offering a non-smooth generalization of catastrophe phenomena.